Identities and inequalities for integral transforms involving squares of the Bessel functions

We consider an integral transform given by Tνf(s):=π∫0∞rsJν(rs)2f(r) dr, where Jν denotes the Bessel function of the first kind of order ν. As shown by Walther (2002), this transform plays an essential role in the study of optimal constants of smoothing estimates for the free Schrödinger equation on Rd. On the other hand, Bez et al. studied these optimal constants using a different method and obtained a certain alternative expression for Tνf involving the d-dimensional Fourier transform of x↦f(|x|) when ν=k+d/2−1 for k∈N. In this paper, we extend their identity to non-integer indices and derive several inequalities from it. In particular, we give a dimension-comparison result for the smoothing estimates on Rd and Rd+1.